Throughout, this paper we work in the smooth category. We assume all 3-manifolds to be connected, closed, oriented, and smooth; all contact structures are positive and coorientable. For background on contact surgery and symplectic and Stein cobordisms we refer to [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. Legendrian links in \((S^3,\xi _{\mathrm {st}})\)  are always presented in their front projection. We choose the normalization of the \(\operatorname{d}_3\) -invariant as in [8]}, [9]} which differs from the normalizations in [12]}, [3]}, [6]} by \(1/2\) . Using our normalization we see that contact structures on homology spheres have integral \(\operatorname{d}_3\) -invariants (in particular \(\operatorname{d}_3(S^3,\xi _{\mathrm {st}})=0\) ) and that the \(\operatorname{d}_3\) -invariant is additive under connected sums.
